1. \( \frac{x^{2}-5 x+6}{2 x-6} \)

To simplify the expression \( \frac{x^{2}-5x+6}{2x-6} \), first factor both the numerator and the denominator. The numerator \( x^2 - 5x + 6 \) can be factored into \( (x - 2)(x - 3) \), and the denominator \( 2x - 6 \) can be factored into \( 2(x - 3) \). This gives us: \[ \frac{(x - 2)(x - 3)}{2(x - 3)} \] Now, we can cancel out the common factor \( (x - 3) \) (as long as \( x \neq 3 \)), resulting in: \[ \frac{x - 2}{2} \] So, the simplified form is: \[ \frac{x - 2}{2} \quad (x \neq 3) \] This simplification helps to avoid any complications when substituting values later on.